Discrete multi-tone (DMT) systems often use a time domain equalizer (TEQ) to shorten the channel and reduce inter symbol interference (ISI), where the TEQ is generally trained using a periodic signal. At the transmit side, the training signal (i.e., the ideal reference) can be generated in the frequency domain as X(k) where k is the bin number. This signal is converted to the time domain via an inverse fast Fourier transform (IFFT) and processed in the later stages by, for example, the digital/analog transmit filter and digital-to-analog converter (DAC) before transmission of data on the channel. At the receiver side, the signal is processed by an analog circuit and sent to an analog-to-digital converter (ADC). Suppose that the time domain signal on the transmit side is x(n) sampled at the Nyquist rate and the channel response is presented by h(n). The received signal after ADC may be determined by linear convolution of the transmit signal and the channel response.
Because the channel length is typically longer than the cyclic prefix length ν, inter-symbol interference (ISI) between consecutive DMT symbols is generated. To reduce ISI, a channel shortening filter or TEQ is applied to the received signal path. Training of the filter W is achieved using a target response filter B which is constrained to a length of ν+1. The TEQ training can be implemented in time domain or in frequency domain. To effectively use the resources (for example, to use the existing fast Fourier transform (FFT) engine for frequency domain signal processing), a frequency domain trained symbol-spaced TEQ (T-spaced TEQ) can be used for channel shortening.
Generally, the advantage of a T-spaced TEQ is its degree of simplicity. However, a well-known problem associated with T-spaced TEQ is the aliasing that folds back into the useful signal band. The transmit filter is typically not sharp enough to filter out the images from the inverse fast Fourier transform (IFFT) stage. A sharper filter requires additional resources (e.g., digital or analog components). Also, a very sharp filter introduces some issues for the TEQ due to the large group delay. Thus, with a T-spaced TEQ, the images above the Nyquist bandwidth will be folded back into the signal band. This aliasing may be added constructively or destructively and therefore, adversely affects performance. While techniques exist for mitigating the effects of aliasing, various perceived shortcomings exist such as divergence of filter coefficients for integer fractionally-spaced TEQ.